7.2.
Spider diffraction

More often than not, central obstruction by a smaller
secondary mirror is accompanied with aperture obstruction caused by its
support structure - so called
spider vanes. Their effect is generally small, but
it can be significant. That makes them worth of a closer look.

Unless the secondary
mirror cell is
supported by an optical window, the supporting vanes - so called spider vanes
- are in the optical
path, altering emitting area of the wavefront and, thus, creating
diffraction effect. As long as the pupil area obstructed by the vanes remains
relatively small, spider diffraction is more of a cosmetic damage than seriously affecting contrast level (FIG.
108). Analogously to the central obstruction effect, what

FIGURE 108: Visual appearance of a bright star without spider
effect (a), with three-vane spider effect (b), and four-vane spider effect
- the two most common spider forms - (c). The
effect is noticeable mainly on objects of high telescopic
brightness.
While the spikes caused by spider vanes can be visually distracting, the amount of energy
lost from the disc is usually negligible for
general observing (3-vane spider spikes are usually shorter,
due to the vanes being generally thicker, as it is needed
for mechanical stability in that spider configuration, but
they may be less intense, since their patterns don't
overlap).

can be thought of as a Strehl ratio degradation factor caused
by spider diffraction is, in effect, the ratio of
the clear (annular) pupil area with and without the vanes, squared, or

with N being the vane count, τ
the relative vane thickness and
ο
the relative size of central obstruction, both in units of the aperture
diameter. The negative factor equals the relative spider area in units
of the clear aperture (i.e. annulus) area; this is consistent with degradation factor
caused by central obstruction (Eq.
60). Average spider area is somewhere between 1% and 2% of
the clear aperture area. That puts an average spider vane contrast
degradation factor between 0.98 and 0.96 - below the level of 1/30
wave RMS wavefront error. As plots above show, it decreases somewhat
with larger obstructions, but that is, of course, paid for with
significantly greater combined Strehl degradation factor due to the
obstruction.

Analogously to the effect of central obstruction, vane
obstruction reduces central intensity of the main pattern by a (1-a)2
factor, a being the relative vane area in units of the clear
aperture area (for spider vanes, it is the area of annulus), by
lowering constructive interference within central maxima and intensifying
it in the outer potion of the pattern. For small values of a, typical
for spider vanes, the
Strehl degradation factor can be written as S'~(1-2a).

It also closely approximates the combined Strehl
degradation factor of the spider and c. obstruction in the left side of
MTF graph (extended low-contrast detail resolution) if a is their combined relative area in the
aperture (this approximation is also good for the PSF maxima degradation
factor for central obstructions smaller than ~0.35D).

This is not quite in agreement with the popular notion
that the contrast effect of spider vanes is directly proportional to
their area, relative to the area of aperture. The misconception probably
comes from misunderstood sequence in Suiter's "Star Testing
Astronomical Telescopes", where he states that the initial quick
contrast drop is in proportion to the vanes area. However, looking at
the MTF graph, it is easy to see that this initial drop in contrast
remains nearly unchanged linearly for nearly 2/3 of the MTF range. In
other words, the actual contrast loss keeps increasing as the relative
contrast value decreases for smaller spatial frequencies (detail size).
The average contrast loss caused by vanes is, therefore, considerably
higher, as given by Eq. 65.

However, it should be emphasized that this theoretical
approach is strictly valid only in the context of near monochromatic
point source - i.e. coherent light. In the real world, light processed
by the telescopes is typically polychromatic, i.e. partly incoherent, in
which case the effect of light obstructions on intensity distribution
within diffraction image is
significantly smaller. As in the section about the effect of central
obstruction, the text continues with the standard coherent light context, but
keeping in mind that it is not directly applicable to the field
conditions.

Depending on the object of observation, the actual spider effect can be much smaller, due to the
energy being thrown so far from the Airy disc. For instance, a spider
wane D/100 thick will have its principal spike length superimposed over
diffraction pattern nearly 100 Airy disc diameters long (only a portion
of it visible at best, depending on its telescopic brightness). For a 10"
aperture, that is nearly 1 arc minute from the disc center. That would
place most of the spike energy out of a relatively small object, not
influencing its contrast. For Jupiter, roughly 2/3 of the principal
spike fall outside the planet's disc, with 1/3, or so, of the spikes' energy
left in, lowering the contrast. Assuming 4-vane spider and 25%
obstruction, it would cause little over 1% actual average contrast loss
(nearly 0.99 Strehl equivalent), not 4% as indicated by Eq. 65. On
the other hand, on large objects like the Moon, nearly entire spikes' energy
remains within the image, and the effective contrast degradation factor
is ~0.96.

There are various vane configurations possible, but the
only result is a different form of energy distribution - the amount of
energy transferred out of the Airy disc remains unchanged for any given vanes area.
Given size of central obstruction, the vane area is directly
proportional to its width - the wider vanes, the more energy spread out,
the higher its peak intensity, but the shorter spike length.

Spider diffraction effect is often illustrated by the
effect of a narrow slit. There doesn't seem to be clearly defined
width above which diffraction effect becomes that of an aperture. Hecht
uses narrow slit relation for a 0.5mm by 30mm slit, which is very
similar to the vane configuration, even if it is, evidently very wide
with respect to the wavelength.
Specifically, intensity distribution within diffraction pattern created by a
slit aperture placed in front of an objective with focal length ƒ
is described by:

with I(0)
being the intensity as a function of point radius, for the central intensity normalized to 1
(actual intensity depends on the slit area), β=Sπsinθ
in units of the wavelength λ, S being the edge-to-edge separation
(i.e. either the slit width, or length)
and θ=r/ƒ being the point angle in the image plane,
in radians, with r the linear point height (linear radius r=β/πi
in units of λƒ/S). The numerator
angle is in degrees, denominator angle in radians. The minimas occur
for β=aπ,
with a=1,2,3,4... First maxima is for β=θ=0, and every subsequent maxima
at β=tanβ (with β at left in radians), or for β=bπ,
with b=1.43, 2.46. 3.47... This gives the second maxima intensity (for
β=1.43π)
as 0.047 of the central intensity, the third maxima as 0.016, the fourth
0.008, and so forth.

With the slit height much larger than
its width, diffracted energy drops to first minima much quicker in the
plane perpendicular to the slit height, negligible in comparison to the
energy spread in the plane perpendicular to the slit's width (for
instance, with the width-to-height ratio of 1:100, sinθ
in
β=Sπsinθ
has to be 100 times smaller in order for β
to have any given value, including the first minima at
β=π.

What seems to be more appropriate
reference shape for the vane in the visual wavelengths is rectangular
aperture. Its width is still much smaller than its height, but it is
quite large relative to the wavelength. In the case of rectangular
aperture, intensity distribution within diffraction pattern is described
with a double squared sinc function:

where subscripts W and H stand for
aperture's width and height, respectively, along its sides, with βW=Wxπ/ƒ
and βH=Hyπ/ƒ,
also in units of λ,
where x and y are the linear point coordinates in image
plane in the horizontal and vertical direction, respectively.
Evidently, this relation gives identical distribution of minimas and
maximas in the two perpendicular planes as Eq. 66 (one
perpendicular to the vane width, the other to vane's height), with
either x or y being the equivalent of r under
Eq. 66, and either being zero along one of the two perpendicular
axes in image plane. Still, this relation is more complete since
directly determining intensity distribution along both perpendicular
axes, and in any chosen direction in the image plane.

With the first intensity minima falling at a constant
nominal value of
β, its angular radius, given by θ=x/ƒ=βWλ/πW
(for small angles sinθ=θ
in radians) is inversely proportional to the width W.
Thus, the longer the vane, the more narrow its spike; the wider vane,
the shorter its spike. A 200x1mm vane - so with W=1 and H=200
(neglecting central obstruction),
for λ=0.00055mm will produce first maxima nearly 4 arc minutes long
(for βW=π) and
about 1.1 arc seconds wide (βH=π); a vane twice as thick will produce maxima half as
long, with its width unchanged. Thicker
vanes may appear to be producing less intrusive, shorter spikes, but
they drain more energy from the Airy disc, causing greater
negative effect on the contrast level.

The actual spike peak diffraction intensity, similarly to
circular aperture, is proportional
to its area, and can be written as I=πΦW/ƒλ2,
where Φ, W,ƒ
are the flux (blocked by the vane), vane width and focal length. The
flux as a product of vane area and flux per unit area, and can be
written as Φ=WHu, u being the flux per unit
area. The W/ƒ
factor reflects the effect of spike size (i.e. length), with the
intensity being proportional to the width for given flux, and
proportional to the square of it considering that the flux also changes
in proportion to the width. Obviously, if we scale a vane to a twice
larger aperture, its area - and flux blocked out - increases fourfold,
and with it its spike intensity. But the same happens with the
aperture's diffraction pattern, whose intensity, I=πΦ/(2λF)2,
also changes with the flux, and their brightness relative to each other
does not change.

If, however, it is the focal length ƒ
that doubles, spike intensity relative to the star diffraction pattern
also doubles, since the latter is spread onto four times larger, and the
former only two times larger area.

The form of pattern change is determined by the vane profile in
the pupil, which in turn determines intensity distribution of the vane
as an aperture. Straight vane projects a spike that is centered over
diffraction pattern, as illustrated on FIG. 109. Since "dark aperture"
created by the vane becomes a part of the wavefront, it projects a spike
centered at the chief ray (i.e. center of the diffraction pattern), extending orthogonally to
the vane orientation, regardless of its orientation in the pupil, or
length (shorter section will produce wider, fainter spike).

FIGURE 109: TOP LEFT: Spider vane diffraction effect is one of a narrow,
elongated rectangular aperture. Its diffraction pattern extends
appropriately less in the direction of elongation, than in the direction orthogonal to
it. The ratio between these two pattern extensions is 1/ς, with ς being the width-to-length ratio of the vane. Consequently, it makes the
central spider vane maxima ~1/ς times wider than the Airy disc of a
telescope. Peak PSF intensities of the vane vs. Airy disc are
proportional to the ratio of their respective areas.
CENTER: Intensity distribution*
within diffraction pattern of 2x200mm opening centered over 200mm ƒ/10
mirror (so ς=0.01), calculated by OSLO. Its central diffraction maxima is nearly 1mm long (purple), and nearly 0.01mm wide (green, magnified
below). At
0.0134mm and 550nm, the Airy disc is only slightly wider than
the spike width, and nearly 80 times higher (blue).
The central vane maxima is ~70 Airy disc diameters long, but how much of it
is visible - if any - depends on star's telescopic brightness and
magnification. Practically all
visual effect of the vane results from its central maxima. With ~80
times (the reciprocal of the area ratio) lower intensity than the
telescope's PSF central maxima (showing intensity distribution over the
Airy disc), it is fainter here than the first bright ring. Its own second
maxima is another 21 times fainter, nearly as faint as the 5th bright ring of
a perfect aperture. However, similarly to the first bright ring,
the central spike appears nearly half as bright as the disc at high
telescopic brightness levels, due to the logarithmic
intensity response of the eye (since its actual intensity is so
much lower, it fades away much sooner with the drop in brightness, as
its intensity approaches visual threshold of detection).
Above, to the right, the high-magnification simulation of the effect of
a 4-vane spider on the diffraction pattern of a
bright star, with the spikes clearly visible, extending far out from the
central disc. Note that, all else equal, the spike width is greater the
greater central obstruction, due to the inverse slit aperture formed by
the vane having its length reduced by a factor (1-o)/2, o
being the relative linear obstruction size (for instance, spike maxima
in the presence of D/3 central obstruction will be three times wider
than that of a vane spanning the entire aperture; with the
combined intensity of its two sections lower by 1/3).
BOTTOM LEFT: PSF of a double slit/vane has double the
central intensity of a single slit/vane of the same width, with the central maxima
of a single vane effectively
fractionalized into 2s-1 maximas (including central maxima), s being the slit/vane separation in units of their (equal)
width W. BOTTOM RIGHT: PSF of a multiple slit/vane has the central
intensity higher by a factor of N than a single slit, N being the slit/vane number.
Within the width of diffraction envelope of a single slit/vane central
maxima, it forms central maxima narrower approximately by a factor
1/5(N-1)s, with (N-2) subsidiary maximas between subsequent
principal maximas (s-th principal maxima is suppressed by a
diffraction envelope null).*Polychromatic intensity
distribution, produced by OSLO, with 25 wavelengths from 440nm to 670nm,
weighed for the spectral response of the eye; weighing may and may not
be appropriate, but the difference is negligible - slightly higher first
minima, and somewhat lower second maxima. Dimensions of the maxima are
somewhat smaller than what monochromatic formula implies (for λ=550nm)
due to polychromatic effect and, perhaps, under-sampling.

Thus a 3-vane spider
forms three spikes centered over diffraction pattern at 120° radially,
while 4-vane spider forms two spikes at a 90° angle. The total energy contained in the pattern is
proportional to the vane area, and so is the peak pattern
intensity. The central maxima of the diffraction spike is approximately
1/ς
times the Airy disc diameter in length. It is approximated by by 2λ/w
angularly (in radians),
λ being the wavelength of light
and w being the vane width.

Since at these small angles there is practically no difference between
the angle and its tangent, linear length of the spike maxima is
approximately 2λƒ/w,
with ƒ
being the mirror focal length; substituting the vane width w in
terms of the aperture diameter as w=ςD
gives the linear spike length as ~2λF/ς,
F being the telescope focal ratio.

As the two insets on the bottom of FIG. 109 show, it is possible
to reduce diffraction effect of a spider vane by replacing a single vane
with two or more parallel vanes. Multiple vane replaces a single central
maxima of a single vane with multiple subsiding maximas covering bearly
identical width angularly. Intensity of the central maxima is
proportional to the combined vane area, thus for the reduction in energy
transferred from the Airy disc, such multiple vane would need to have
unit vanes of lesser width than a single vane it would replace.

Curved vane spider

The intense spike produced by a straight vane can be
visually eliminated by
curving the vanes. The result is a
curved vane spider. Diffraction effect of a curved vane can be
illustrated by breaking it into a number of
smaller, practically straight sections, with varying orientations (FIG.
110a). While the total
amount of energy produced by a curved vane is identical to that of a
straight vane of equal length and thickness, it is spread out wide,
making it practically invisible (it still lowers the contrast the same,
on average).

The Strehl degradation factor is somewhat different from
that for the straight vanes (Eq. 65):

with α=180/N being the vane arch angle
in degrees.
However, the result is only slightly lower for given count (N) and
relative thickness (τ) of the vanes, reflecting
the slightly
greater curved vane length.

FIGURE 110: Diffraction effect of a curved spider
vane: (a) Each practically straight vane section produces an
effective slit at the aperture, which projects its diffraction pattern
onto the main pattern (bottom). The patterns of the individual sections
are wider, due to their reduced length, but their extension is unchanged
(it only varies with the vane width).
The orientation of each of these patterns (1,2,3
and 4) is orthogonal to that of the vane section, with the total angle
of pattern spread being approximately that of the radial angle of the
vane arch α (centered on its radius
of curvature). Note that the actual effectively straight sections of a
curved vane are shorter than those shown on the illustration, and the pattern spread
is continuous within the spread angle. To spread the energy as
much as possible - over 360° angle - and, at the same time, avoid overlapping, the vanes
need to be appropriately curved and positioned. The simple rule is that
a spider with N vanes needs to have 180/N vane arch angle (in
degrees), with every next vane separated radially either by the same (180/N
degrees) angle, or by (180/N)+180 degrees angle, whichever is
mechanically superior (to which side the vane is curved is
irrelevant). Thus, a 4-vane spider
(b) requires
α=45°, and either 45° or 225° between the
vanes (B' and C' are alternative positions for better
stability, with the D vane preferably rotated length-wise). For a
3-vane curved spider (c),the arch angle is 60°, with the
second vane separated from the first one by either 60° or 240°, and the
third vane by as much from the second vane, clockwise; B' is the alternative
B vane position. A 2-vane curved spider
(d) needs 90° arch
angle, with the second vane at either 90° or 270° from the first one (B'
are alternative positions for the second vane). A
single curved vane (e) would require 180° vane arch; hence more
stable, circular form would double the energy spread out,
which still would be at the level of a 2-vane spider.